Step |
Hyp |
Ref |
Expression |
1 |
|
hbtlem.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
2 |
|
hbtlem.u |
⊢ 𝑈 = ( LIdeal ‘ 𝑃 ) |
3 |
|
hbtlem.s |
⊢ 𝑆 = ( ldgIdlSeq ‘ 𝑅 ) |
4 |
|
hbtlem.d |
⊢ 𝐷 = ( deg1 ‘ 𝑅 ) |
5 |
|
elex |
⊢ ( 𝑅 ∈ 𝑉 → 𝑅 ∈ V ) |
6 |
|
fveq2 |
⊢ ( 𝑟 = 𝑅 → ( Poly1 ‘ 𝑟 ) = ( Poly1 ‘ 𝑅 ) ) |
7 |
6 1
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ( Poly1 ‘ 𝑟 ) = 𝑃 ) |
8 |
7
|
fveq2d |
⊢ ( 𝑟 = 𝑅 → ( LIdeal ‘ ( Poly1 ‘ 𝑟 ) ) = ( LIdeal ‘ 𝑃 ) ) |
9 |
8 2
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ( LIdeal ‘ ( Poly1 ‘ 𝑟 ) ) = 𝑈 ) |
10 |
|
fveq2 |
⊢ ( 𝑟 = 𝑅 → ( deg1 ‘ 𝑟 ) = ( deg1 ‘ 𝑅 ) ) |
11 |
10 4
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ( deg1 ‘ 𝑟 ) = 𝐷 ) |
12 |
11
|
fveq1d |
⊢ ( 𝑟 = 𝑅 → ( ( deg1 ‘ 𝑟 ) ‘ 𝑘 ) = ( 𝐷 ‘ 𝑘 ) ) |
13 |
12
|
breq1d |
⊢ ( 𝑟 = 𝑅 → ( ( ( deg1 ‘ 𝑟 ) ‘ 𝑘 ) ≤ 𝑥 ↔ ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ) ) |
14 |
13
|
anbi1d |
⊢ ( 𝑟 = 𝑅 → ( ( ( ( deg1 ‘ 𝑟 ) ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) ↔ ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) ) ) |
15 |
14
|
rexbidv |
⊢ ( 𝑟 = 𝑅 → ( ∃ 𝑘 ∈ 𝑖 ( ( ( deg1 ‘ 𝑟 ) ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) ↔ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) ) ) |
16 |
15
|
abbidv |
⊢ ( 𝑟 = 𝑅 → { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( ( deg1 ‘ 𝑟 ) ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } = { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) |
17 |
16
|
mpteq2dv |
⊢ ( 𝑟 = 𝑅 → ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( ( deg1 ‘ 𝑟 ) ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) = ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) |
18 |
9 17
|
mpteq12dv |
⊢ ( 𝑟 = 𝑅 → ( 𝑖 ∈ ( LIdeal ‘ ( Poly1 ‘ 𝑟 ) ) ↦ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( ( deg1 ‘ 𝑟 ) ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) = ( 𝑖 ∈ 𝑈 ↦ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) ) |
19 |
|
df-ldgis |
⊢ ldgIdlSeq = ( 𝑟 ∈ V ↦ ( 𝑖 ∈ ( LIdeal ‘ ( Poly1 ‘ 𝑟 ) ) ↦ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( ( deg1 ‘ 𝑟 ) ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) ) |
20 |
18 19 2
|
mptfvmpt |
⊢ ( 𝑅 ∈ V → ( ldgIdlSeq ‘ 𝑅 ) = ( 𝑖 ∈ 𝑈 ↦ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) ) |
21 |
5 20
|
syl |
⊢ ( 𝑅 ∈ 𝑉 → ( ldgIdlSeq ‘ 𝑅 ) = ( 𝑖 ∈ 𝑈 ↦ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) ) |
22 |
3 21
|
syl5eq |
⊢ ( 𝑅 ∈ 𝑉 → 𝑆 = ( 𝑖 ∈ 𝑈 ↦ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) ) |
23 |
22
|
fveq1d |
⊢ ( 𝑅 ∈ 𝑉 → ( 𝑆 ‘ 𝐼 ) = ( ( 𝑖 ∈ 𝑈 ↦ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) ‘ 𝐼 ) ) |
24 |
23
|
fveq1d |
⊢ ( 𝑅 ∈ 𝑉 → ( ( 𝑆 ‘ 𝐼 ) ‘ 𝑋 ) = ( ( ( 𝑖 ∈ 𝑈 ↦ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) ‘ 𝐼 ) ‘ 𝑋 ) ) |
25 |
24
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0 ) → ( ( 𝑆 ‘ 𝐼 ) ‘ 𝑋 ) = ( ( ( 𝑖 ∈ 𝑈 ↦ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) ‘ 𝐼 ) ‘ 𝑋 ) ) |
26 |
|
rexeq |
⊢ ( 𝑖 = 𝐼 → ( ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) ↔ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) ) ) |
27 |
26
|
abbidv |
⊢ ( 𝑖 = 𝐼 → { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } = { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) |
28 |
27
|
mpteq2dv |
⊢ ( 𝑖 = 𝐼 → ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) = ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) |
29 |
|
eqid |
⊢ ( 𝑖 ∈ 𝑈 ↦ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) = ( 𝑖 ∈ 𝑈 ↦ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) |
30 |
|
nn0ex |
⊢ ℕ0 ∈ V |
31 |
30
|
mptex |
⊢ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ∈ V |
32 |
28 29 31
|
fvmpt |
⊢ ( 𝐼 ∈ 𝑈 → ( ( 𝑖 ∈ 𝑈 ↦ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) ‘ 𝐼 ) = ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) |
33 |
32
|
fveq1d |
⊢ ( 𝐼 ∈ 𝑈 → ( ( ( 𝑖 ∈ 𝑈 ↦ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) ‘ 𝐼 ) ‘ 𝑋 ) = ( ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ‘ 𝑋 ) ) |
34 |
33
|
3ad2ant2 |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0 ) → ( ( ( 𝑖 ∈ 𝑈 ↦ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝑖 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ) ‘ 𝐼 ) ‘ 𝑋 ) = ( ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ‘ 𝑋 ) ) |
35 |
|
eqid |
⊢ ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) = ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) |
36 |
|
breq2 |
⊢ ( 𝑥 = 𝑋 → ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ↔ ( 𝐷 ‘ 𝑘 ) ≤ 𝑋 ) ) |
37 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) ) |
38 |
37
|
eqeq2d |
⊢ ( 𝑥 = 𝑋 → ( 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ↔ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) ) ) |
39 |
36 38
|
anbi12d |
⊢ ( 𝑥 = 𝑋 → ( ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) ↔ ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑋 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) ) ) ) |
40 |
39
|
rexbidv |
⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) ↔ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑋 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) ) ) ) |
41 |
40
|
abbidv |
⊢ ( 𝑥 = 𝑋 → { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } = { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑋 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) ) } ) |
42 |
|
simp3 |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0 ) → 𝑋 ∈ ℕ0 ) |
43 |
|
simpr |
⊢ ( ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑋 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) ) → 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) ) |
44 |
43
|
reximi |
⊢ ( ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑋 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) ) → ∃ 𝑘 ∈ 𝐼 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) ) |
45 |
44
|
ss2abi |
⊢ { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑋 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) ) } ⊆ { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) } |
46 |
|
abrexexg |
⊢ ( 𝐼 ∈ 𝑈 → { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) } ∈ V ) |
47 |
46
|
3ad2ant2 |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0 ) → { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) } ∈ V ) |
48 |
|
ssexg |
⊢ ( ( { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑋 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) ) } ⊆ { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) } ∧ { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) } ∈ V ) → { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑋 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) ) } ∈ V ) |
49 |
45 47 48
|
sylancr |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0 ) → { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑋 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) ) } ∈ V ) |
50 |
35 41 42 49
|
fvmptd3 |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0 ) → ( ( 𝑥 ∈ ℕ0 ↦ { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑥 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑥 ) ) } ) ‘ 𝑋 ) = { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑋 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) ) } ) |
51 |
25 34 50
|
3eqtrd |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0 ) → ( ( 𝑆 ‘ 𝐼 ) ‘ 𝑋 ) = { 𝑗 ∣ ∃ 𝑘 ∈ 𝐼 ( ( 𝐷 ‘ 𝑘 ) ≤ 𝑋 ∧ 𝑗 = ( ( coe1 ‘ 𝑘 ) ‘ 𝑋 ) ) } ) |